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Theorem eldmressn 39852
 Description: Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Assertion
Ref Expression
eldmressn (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴)

Proof of Theorem eldmressn
StepHypRef Expression
1 elin 3758 . . 3 (𝐵 ∈ ({𝐴} ∩ dom 𝐹) ↔ (𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹))
2 elsni 4142 . . . 4 (𝐵 ∈ {𝐴} → 𝐵 = 𝐴)
32adantr 480 . . 3 ((𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹) → 𝐵 = 𝐴)
41, 3sylbi 206 . 2 (𝐵 ∈ ({𝐴} ∩ dom 𝐹) → 𝐵 = 𝐴)
5 dmres 5339 . 2 dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹)
64, 5eleq2s 2706 1 (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  {csn 4125  dom cdm 5038   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-res 5050 This theorem is referenced by:  dfdfat2  39860
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